theorem
  [.a,b.]" = [.a,b".]|^b
proof
  thus [.a,b.]" = ((a" * b") * (a * b))" by GROUP_1:def 3
          .= (a * b)" * (a" * b")" by GROUP_1:17
          .= (b" * a") * (a" * b")" by GROUP_1:17
          .= (b" * a") * (b"" * a"") by GROUP_1:17
          .= b" * (a" * (b * a)) by GROUP_1:def 3
          .= b" * (a" * b * a) by GROUP_1:def 3
          .= b" * (a" * b * a) * 1_G by GROUP_1:def 4
          .= b" * (a" * b * a) * (b" * b) by GROUP_1:def 5
          .= b" * (a" * b * a) * b" * b by GROUP_1:def 3
          .= [.a,b".]|^b by GROUP_1:def 3;
end;
